Some notation for vector space $\mathbb{R}^\mathbb{R}$, $\mathbb{R}^{X}$, $C(X)$ I am reading some slides for functional analysis, and it mentioned that $\mathbb{R}^\mathbb{R}$, $\mathbb{R}^{X}$, and $C(X)$ are all vector spaces. Since the slides are so brief and it doesn't provide an further details. Is there anyone can provide me some definition of these notation?
My guess is that $\mathbb{R}^\mathbb{R}$ is all the function defined on real numbers. $C(X)$ might be related to continuous functions. 
 A: For sets $X$ and $Y$, the set of maps $X \to Y$ is sometimes denoted $Y^X$. (Compare the power set $2^X$, which can be thought of as the set of maps $X \to \{0, 1\}$ by associating to a set $U\subset X$ its indicator function.) If $Y$ has some additional structure, then $Y^X$ generally does as well; in particular, if $Y$ is a vector space, then $Y^X$ has a vector space structure with operations $(f+g)(x) = f(x) + g(x)$ and $(\lambda f)(x) = \lambda f(x)$. (Note that we're not taking $Y^X$ to be the space of linear maps $X \to Y$, even if $X$ is also a vector space.) In functional analysis, you probably want to restrict $Y^X$ to the space of continuous maps $X \to Y$; arbitrary functions aren't very interesting.
For a topological space $X$, the space of continuous maps $X \to \mathbb{R}$ is sometimes denoted by $C(X)$ or $C(X, \mathbb{R})$; the complex case is analogous. It's also common to write $C_0(X)$ for compactly supported functions $X \to \mathbb{R}$ or $X \to \mathbb{C}$, along with $C^p(X), C^\infty(X),$ and $C^\omega(X)$ for elements of $C(X)$ that have $p$ continuous derivatives, are smooth, and are analytic, respectively.
